Dispersion shifted optical waveguide fiber

ABSTRACT

A single mode optical waveguide fiber designed for high data rate, or WDM systems or systems incorporating optical amplifiers. The optical waveguide has a compound core having a central region and at least one annular region surrounding the central region. A distinguishing feature of the waveguide core is that the minimum refractive index of the central core region is less than the minimum index of the adjacent annular region. A relatively simple profile design has the characteristics of ease in manufacturing together with, flexibility in tailoring D w  to yield a preselected zero dispersion wavelength, dispersion magnitude over a target wavelength range, and dispersion slope. The simplicity of profile gives reduced polarization mode dispersion.

BACKGROUND

The invention is directed to a single mode optical waveguide fiberwherein a refractive index profile design is optimized for high datarate links, or systems using optical amplifiers, or wavelength divisionmultiplexed systems.

The full capability of optical waveguide fiber is being exploited byhigh data rate systems having a long distance between repeaters. Theoperating window in a range including 1550 nm is attractive for thesesystems because of the lower attenuation possible and the absence ofabsorption peaks. Data rates typical of such systems are greater than 1gigabit/sec and repeater spacing exceeds 50 km.

The high data rates require that the birefringence of the waveguidefiber be low. That is, the dispersion of the polarizations of the singlepropagated mode must be controlled to limit bit errors. The high datarates also require that the zero dispersion wavelength be near 1550 nmto limit material dispersion. Furthermore, the introduction of highpowered lasers has produced non-linear effects which can limit data rateor repeater spacing. In systems which utilize wavelength divisionmultiplexing over a relatively small wavelength range, the non-linearinterference effect called four wave mixing (FWM) is especiallydetrimental.

One approach to limiting polarization mode dispersion (PMD) is toprovide a waveguide fiber which is relatively free of birefringence.This may be accomplished by maintaining circularly symmetric geometryand by limiting residual stress in the fiber. In addition, a waveguidehaving a relatively lower dopant level in the signal carrying portion ofthe waveguide will have reduced Rayleigh scattering and will reduce biterrors due to non-linear effects.

The impact of non-linear effects can also be lessened by providing alarger mode field diameter to reduce power density in the waveguidefiber. Four wave mixing can essentially be eliminated by moving the zerodispersion wavelength out of the operating window. A non-zero dispersionover the operating window serves to prevent the phase matching ofmultiplexed signals thereby eliminating the four wave mixing signalinterference.

The objectives, therefore, in manufacturing a waveguide fiber for highdata rate, long repeater spacing and multichannel operation are toprovide:

low residual stress;

reduced overlap of signal with higher dopant waveguide regions;

higher modefield; and,

dispersion zero away from the operating window.

Further, these properties must be achieved while maintaining lowattenuation, acceptable bend performance and appropriate cut offwavelength. An added benefit can be realized if the performance goalscan be met without increasing manufacturing difficulty or cost.

SUMMARY OF THE INVENTION

The present invention fulfills the requirements for a high performancewaveguide fiber. Further, a waveguide of the inventive design isrelatively easier to manufacture and thus is lower in manufacturingcost.

A major feature of this invention, which distinguishes it from othercompound core profile designs, is that a central core region ismaintained relatively low in dopant content. This central region isadjacent to a region relatively higher in dopant content. Theadvantageous result is a profile design flexible enough to satisfy anexacting specification but simple enough to allow ease of manufactureusing standard equipment. The inventive profile effectively controlsindex on centerline and moves the index peak to an off centerlineposition.

A first aspect of the invention is a single mode optical waveguide fiberhaving a compound core. A central region of the core has a minimumrefractive index n₀ and a radius a₀. The central core region issurrounded by at least one annular core region where the innermost ofthe annular regions has a minimum refractive index n_(i) and a radiusa_(i) and where n_(i) >n₀ and a_(i) >a₀. The core is surrounded by acladding layer having refractive index n_(c) where n_(i) >n_(c). Thehighest index point of the central core region may occur at or near thewaveguide axial centerline.

In general, the refractive index of the central region and therefractive indices of the at least one surrounding annular region mayvary with radius. A preferred embodiment of the inventive refractiveindex profile is one in which the refractive index in each core regionis essentially cylindrically symmetrical.

In another preferred embodiment, the waveguide profile is essentiallycylindrically symmetric and the core comprises one annular regionsurrounding the central core region.

A most preferred embodiment has a cylindrically symmetric waveguiderefractive index and a core refractive index profile including asubstantially constant index over a single annular region surroundingthe central core region. The central core region index may also besubstantially constant in this embodiment. Further, the central coreregion index may be substantially equal to the refractive index of thecladding, i.e., the central core % delta is inside the range +/-0.1%.

Also contemplated are designs which reduce the refractive index,relative to the refractive index of silica, of all or part of any of thecore regions or all or part of the clad layer.

Another aspect of the invention is a waveguide fiber having a centralcore surrounded by two annular regions having respective minimumrefractive indices n₁ and n₂. The first annular region is adjacent thecentral core region and the second annular region surrounds and isadjacent to the first annular region. The relationship of the refractiveindices of the respective regions is n₁ >n₀ and n₂ >n₀, where n₀ is thecentral core region minimum refractive index.

A further aspect of the invention is a single mode optical waveguidefiber including a central core region having a substantially constantrefractive index n₀. The central core region is surrounded by at leastone annular region. The annular region adjacent the core has minimumrefractive index n_(i), where n_(i) >n₀. The waveguide has a clad layerhaving refractive index n_(c) surrounding the core region.

In a preferred embodiment, the substantially constant refractive indexof the central core region is substantially equal to the refractiveindex of the clad layer. In this embodiment the total dispersion slopecan be less than about 0.05 ps/nm² /km. The maximum dispersion slope ofthis embodiment is typically no greater than 0.075 ps/nm² /km. Theembodiment is relatively free of draw induced residual stress and stressdue to thermal expansion mismatch. In addition, the zero dispersionwavelength is relatively insensitive to changes in cut off wavelength orcore diameter. A change of about 5% in either cut off wavelength or corediameter produces substantially no change in zero dispersion wavelength.Furthermore, in this embodiment the zero dispersion wavelength can bemoved away from the operating wavelength range to a wavelength less thanabout 1530 nm or greater than about 1565 nm.

Yet another aspect of the invention is a single mode optical waveguidefiber including a core having an axially symmetric central region ofminimum refractive index n₀ surrounded by an axially symmetric annularregion of minimum refractive index n₁, an inner radius a_(i) and anouter radius a_(o), where n₁ >n₀ and the ratio a_(i) /a_(o) is in therange of about 0.35 to 0.80. The core is surrounded by a clad layer ofrefractive index n_(c), where n₁ >n_(c).

In preferred embodiments of this aspect, n₀ is substantially constant,or n₀ is substantially equal to n_(c), or n₁ is substantially constant.A preferred value of n₁ is in the range of about 1.4700 to 1.4800.

Other features and advantages of the inventive refractive index profilewill be described in the detailed description in conjunction with thefollowing drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an example of compound core optical waveguide fiber prior art.

FIGS. 2a-e are illustrations of several embodiments of the inventiverefractive index profiles charted versus radial position in thewaveguide.

FIG. 3 is an illustration of an embodiment wherein a part of the centralcore region and a part of the clad layer have a refractive index lessthan the index of silica.

FIG. 4 is an illustrative charts showing three alternative waveguidedispersion curves possible using the inventive profile.

FIG. 5 is an illustrative chart showing an example of the dependence ofzero dispersion wavelength on the waveguide core radius.

FIG. 6 is an illustrative chart showing an example of total dispersion,material dispersion and waveguide dispersion.

FIG. 7 is an illustrative chart showing an example of the signal fieldamplitude relative to the index profile of an embodiment of theinventive profile.

FIG. 8 is a chart of an actual example of the inventive index profile.

DETAILED DESCRIPTION OF THE INVENTION

The drawings are intended to aid in understanding the invention and inno way limit the invention. The drawings are not necessarily to scale.

The terms refractive index and index profile and index are usedinterchangeably.

The radii of successive regions of the core are defined in terms ofindex of refraction. Thus the central core region has a radius which isdefined as the distance from the core center to a point on the corediameter whereat the refractive index has a prescribed valuecharacterizing the end of the central region. The inner and outer radiiof annular core regions are defined analogously. For example, the innerradius of an annular core region is the radius at which the refractiveindex has a prescribed value characterizing the beginning of an annularregion.

An early example of the use of a compound core design to provide awaveguide which meets a wide range of specifications is found in U.S.Pat. No. 4,715,679, Bhagavatula, incorporated herein by reference. The'679 patent shows how the introduction of a plurality of core regionshaving various dimension and refractive index provides the flexibilityto construct a waveguide having a particular waveguide dispersion. Asdefined in the '679 patent, the total dispersion, D_(t), is thealgebraic sum of the material dispersion, D_(m), and the waveguidedispersion, D_(w). A waveguide can be tailored to meet a specified setof properties including, cut off wavelength, zero dispersion wavelength,and mode field diameter.

An example of a compound profile of the '679 patent is shown in theindex versus radial position chart in FIG. 1. The center region 2 issurrounded by adjacent region 4 wherein region 4 in general has a lowerrefractive index than region 1. The remainder of the core is comprisedof successive regions 6, 8, and 10. The refractive index profile in therespective regions may have essentially any shape. The dashed lines inregions 4 and 8 indicate the refractive index is not limited to aparticular value. The core is surrounded by a cladding layer 14 whichmay have a depressed region 12.

In contrast to this art is the inventive compound core profile, oneembodiment of which is illustrated in FIG. 2a. In this embodiment, thecentral region of the core, delimited by radius 22, has a refractiveindex profile 16 of general shape with minimum point 18. The annularregion of the core adjacent the central region has an index profile 20of general shape. The dashed line 21 indicates an alternative indexprofile for the annular region. No point of index 20 is less thanminimum point 18. Even with this distinguishing limitation imposed, ithas been found possible to devise index profiles which tailor waveguidedispersion and so provide for flexibility in designing an opticalwaveguide having a particular set of characteristics.

In another embodiment, the index profile in the central core region issubstantially constant as shown by 24 in FIG. 2b. The adjacent annularregion has index profile 26. The dashed lines in the annular region ofthe profile in FIG. 2b indicate the index profile 26 may vary from pointto point along the radius. The waveguide fiber made in accord with theindex profile of FIG. 2b is preferred, in terms of the ease ofmanufacture, providing the pertinent specification can be met using sucha profile.

A further simplification of the inventive profile is illustrated in FIG.2c. In this embodiment, the central region 28 is substantially constantand substantially equal to the refractive index of the clad layer. Theannular region adjacent the central core region may have a general shapewhich varies with radial position.

A waveguide fiber made in accord with the profile of FIG. 2c ispreferred because:

the index profile is readily manufactured using standard equipment;

the uniformity of composition (only one annular region contains adopant) results in less waveguide stress due to thermal mismatch; and,

the simplicity of the design can result in improved circularity andconcentricity of the core regions and the clad layer. The ease ofmanufacture translates directly to cost reduction. The reduced thermalmismatch stress and the improved geometry translate directly intoreduced polarization mode dispersion.

In addition to these benefits, the core still has sufficient flexibilityto allow for tailoring of the waveguide dispersion, D_(w), to meet awide range of waveguide applications. In particular, a waveguide fibermade in accord with the index profile of FIG. 2c has been tailored toexhibit the properties required for high data rate systems which employwavelength division multiplexing or which require long repeater spacingor which use optical amplifiers.

EXAMPLE

A single mode optical waveguide fiber was manufactured having an indexprofile similar to that shown in FIG. 2c. The waveguide had a singleannular core region of inner radius 0.93 and outer radius 1.9. The indexdelta of the annual region was about 0.9%, where,

index delta=(n_(max) ² -n_(c) ²)/2n_(max) ², and n_(max) is the maximumindex of the annular core region. The index profile of the annulus wasessentially a step with the top and bottom corners rounded due todiffusion of the GeO₂ dopant. The profile is shown as curve 68 in FIG.8. Note that the inner and outer radius, points 70 and 72, are found bydrawing perpendicular lines to the x-axis from about the half maximumindex points.

The central core region had a substantially constant index profile ofindex delta 0.1%. The cladding layer was SiO₂.

The waveguide fiber properties were:

mode field diameter--8.2 microns;

dispersion zero--1593 nm;

dispersion slope--0.044 ps/km-nm² ; and,

PMD--0.031 ps/km^(1/2).

Note that these properties meet the requirements, known in the art, forhigh data rate systems, or WDM systems, or systems using opticalamplifiers.

Using the example profile design, it was found that the zero dispersionwavelength could be shifted to essentially any point in the wavelengthrange from about 1475 to 1600 nm.

                  TABLE I                                                         ______________________________________                                        a.sub.i Disp. Zero                                                                              Slope      Cut Off                                                                              PMD                                       ______________________________________                                        1.10    1486      0.062      1099   0.055                                     1.06    1535      0.073      1000   0.088                                     0.93    1593      0.439       929   0.031                                     ______________________________________                                    

Table I. is a comparison of properties of two additional waveguidefibers essentially identical to the waveguide of example 1 except forthe inner radius a_(i) and outer radius a₀ of the annulus. The ratio,a_(i) /a_(o), is the same for the three waveguides. The last row of thetable is the waveguide of example 1. The trends in the data for zerodispersion wavelength and cut off wavelength versus a_(i) are clear. Thetable also shows that total dispersion slope and polarization modedispersion depend upon more than just the placement of a_(i) and a_(o)as is discussed below.

Additional embodiments of the inventive profile are shown in FIGS. 2d &2e and in FIG. 3. FIG. 2d has a central core region 30 wherein the indexcan vary with radius. Two annular regions, 32 and 34 respectively,surround the central region and in general may vary with radius. Thedashed lines indicate that any of the regions may be constant or have adifferent shape from that shown. That is, the distinguishing feature isthat the maximum central core index is less than the minimum index inthe annular core region adjacent the central region.

FIG. 2e shows the embodiment wherein there are several successiveannular regions, 36, 37, 38, and 39. The number of such regions islimited only by the practical considerations of minimum annularthickness required to interact with a light signal and the maximum sizeof the core region in view of the target cut off wavelength.

A further generalization of the inventive waveguide fiber includesregions wherein the refractive index is lower than the index of silica.FIG. 3 shows a single annulus 42 and regions 40 and 44 having an indexbelow that of silica. Such an index profile may be less desirable fromthe standpoint of cost and ease of manufacture.

It is advantageous, both from the standpoint of cost and ease ofmanufacturing to specification, that a simple embodiment of theinventive profile allows appropriate tailoring of the waveguidedispersion. FIG. 4 shows three waveguide dispersion curves possibleusing the embodiment illustrated in FIG. 2c. Curve 46 is the calculatedD_(w) for this embodiment wherein the ratio of inner to outer radius ofthe annulus is in the range of about 0.35 to 0.45. Curve 48, which showssubstantially linear regions at shorter and longer wavelengths separatedby a transition region at intermediate wavelengths, is modelled using aratio of about 0.5. Curve 50, which is substantially constant over awide wavelength range, was calculated using a ratio in the range ofabout 0.55 to 0.75. Other D_(w) curves may be obtained by choosing theappropriate values for the variables, inner radius, outer radius, andindex profile in the central core and annular core regions. Inparticular, D_(w) curves having a concavity opposite to that of curve 46may be achieved.

Because the material dispersion, D_(m), does not change rapidly withchanges in the waveguide variables noted immediately above, changes inthe shape of the D_(w) curve produce corresponding changes in totaldispersion D_(t). D_(m) is taken to be positive by generally acceptedconvention. And D_(w) is therefore negative because it has an oppositeeffect on signal velocity relative to D_(m). Thus curve 46 will producea D_(t) which has a low slope, of the order of about 0.05 ps/nm² -km,over a particular wavelength range, because the material and waveguidedispersion will substantially cancel each other over that range. Thisdesign gives a dispersion zero which will change more rapidly withchanges in core diameter and cut off wavelength compared to the designwhich produces curve 50. In this latter case, the slope of totaldispersion is increased, to about 0.075 ps/nm² -km, so that the zero ofD_(t) varies less as core diameter or cut off wavelength vary. Apreferred slope range for ease of manufacture while maintaining lowdispersion over a range of wavelengths is about 0.055 to 0.060 ps/nm²-km. Note that different parts of the D_(w) curve may have differentslopes, as is shown in curve 48. The different slopes may be located indifferent wavelength regions. Thus a wide range of magnitudes and shapesof D_(t) may be achieved. In general, the design may be optimized tomeet a targeted balance between ease of manufacture and the magnitude ofthe total dispersion over a selected wavelength range.

FIG. 6 shows a representative D_(m) curve 58 and a particular D_(w)curve 60 using the single annulus embodiment of the inventive indexprofile. The resulting D_(t) curve 62 has a characteristic region 64,which includes a wavelength range from about 1520 nm to 1600 nm, whereinthe total dispersion is in the range of about 0.5 ps/nm-km to 4ps/nm-km. This characteristic is ideal for high data rate, wavelengthdivision multiplexing (WDM) in the 1550 nm operating window, becausedispersion is low enough for high data rate but still sufficient toprevent FWM.

Note that the dispersion zero in FIG. 6 is near 1500 nm. The inventiveprofile can be used to place dispersion zero at another targetwavelength, viz., 1565 nm or greater. For example, dispersion zero for ahigh data rate submarine system may be chosen to be in about a 10 nmrange about 5 nm above or below the gain peak of an optical amplifier.Typically this gain peak is near 1558 nm. Thus an optimum choice tolimit non-linear effects and allow optimum use of optical amplifierspacing would be dispersion zero near but not at the amplifier gainpeak, for example, in the range of about 1545 nm to 1555 nm or about1560 nm to 1570 nm. A waveguide so designed is particularly suited foruse in a long length system which does not employ WDM, such as a singlewavelength submarine system. The zero dispersion wavelength of theinventive optical waveguide can be placed at essentially any desiredwavelength.

Theoretical studies of the inventive profile show that choosing zerodispersion in the range of about 1500 nm to 1530 nm is preferred forhigh rate WDM systems because:

a total dispersion slope of 0.05 ps/nm² -km can be achieved;

manufacturing is relatively easier;

bend performance is better; and,

total dispersion over a range of about 1540 nm to 1560 nm is in therange of about 0.5 to 2.5 ps/nm-km. Further, the total dispersion ispositive, according to the sign convention noted above, a conditionessential for soliton systems.

It will be understood that although the examples deal primarily with the1550 nm window, the concept of using the inventive index to tailor D_(w)to obtain a target D_(t) curve may be extended to include operatingwindows above or below the 1550 nm window. For example it iscontemplated that through use of suitable index altering dopants, awaveguide capable of high rate, WDM operation at 1300 nm or atwavelengths substantially above 1550 nm can be designed.

FIG. 5 serves to illustrate the flexibility of the single annulusembodiment of the inventive profile in terms of placement of zerodispersion wavelength. The core radius, i.e., the outer radius of theannulus, is shown on the x-axis. The zero dispersion wavelength is shownon the y-axis. In the relatively low slope part of the curve, 52, zerodispersion wavelength varies from about 1570 nm to 1535 nm as radiusvaries from 3 microns to about 3.4 microns. The part of the curvelabelled 56 is also low slope and zero dispersion varies between about1470 nm and 1455 nm as radius varies from about 3.5 to 3.9 microns. Forthe particular embodiment of FIG. 5, a radius between 3.4 and 3.5microns, segment 54, would give wide variations in zero dispersionwavelength for relatively small changes in radius. This region could beused to effectively randomize dispersion zero and thereby manage totaldispersion within a waveguide system.

The single or multiple annulus core profile design will have reducedpolarization mode dispersion because:

the design is simple and therefore geometry tolerances will berelatively easy to control in manufacturing, i.e., core and cladroundness and concentricity will be improved;

those designs which have low dopant level in the central region of thecore have relatively less overlap between the field of the propagatingsignal and the high dopant core region.

For a single annulus design the power distribution of the signalrelative to the index profile is shown in FIG. 7. The index profile ofthe waveguide is illustrated by curve 66. A family of field intensitycurves, 64, are plotted on the same chart. Each member of the family ofcurves has a different V value. The curves show power density is highestin the core region having the lowest dopant level, i.e., the regionwhere polarization birefringence is least likely to occur. Thecoincidence of high power density with lowest dopant level also producesthe advantages of lower Rayleigh scattering and lower impact ofnon-linear effects, such as self phase modulation or FWM.

Thus the inventive refractive index profile, in its several embodiments,offers flexibility of design together with ease of manufacture. Inparticular, the inventive profile may be chosen:

to select a particular range for the zero dispersion wavelength;

the zero dispersion wavelength may be made relatively insensitive tomanufacturing variations in core radius or cut off wavelength;

the profile may be optimized for high data rate, WDM systems; and,

the profile may be optimized for use with optical amplifiers, especiallywhere spacing between repeaters is large.

Although specific embodiments of my invention have hereinbefore beendisclosed and described, it will be understood that the scope of myinvention is nevertheless to be defined by the following claims.

I claim:
 1. A single mode optical waveguide fiber comprising:a coreregion comprising, a central region having a radius a₀ and a minimumrefractive index n₀, at least one annular region surrounding saidcentral region, wherein the innermost of said at least one annularregion is adjacent said central region and has inner radius a_(i) andminimum refractive index n_(i), and a_(i) >a₀ and n_(i) >n₀ ; and, aclad layer adjacent said core region having refractive index n_(c) andn_(c) <n_(i).
 2. The single mode optical waveguide of claim 1 whereinthe respective refractive indices of said central region and said atleast one annular region vary along the radius of said core.
 3. Thesingle mode optical waveguide of claim 2 wherein the respectiverefractive indices of said central region and said at least one annularregion of said core are cylindrically symmetrical.
 4. The single modeoptical waveguide fiber of claim 1 wherein said central core region issurrounded by one adjacent annular region.
 5. The single mode opticalwaveguide fiber of claim 4 wherein said one annular region has asubstantially constant refractive index.
 6. The single mode opticalwaveguide fiber of claim 5 wherein said central region has asubstantially constant refractive index.
 7. The single mode opticalwaveguide fiber of claim 6 wherein n₀ is substantially equal to n_(c).8. The single mode waveguide fiber of claim 1 further including at leastone annular region in said clad layer having a refractive index lessthan n_(c).
 9. The single mode waveguide fiber of claim 1 furtherincluding at least one annular region in said central core region havinga refractive index less than n_(c).
 10. The single mode opticalwaveguide fiber of claim 1 wherein said central core region issurrounded by a first and a second annular region, wherein said firstannular region is adjacent said central region and said second annularregion is adjacent said first annular region.
 11. The single modeoptical waveguide of claim 10 wherein said respective first and secondannular regions have minimum refractive indices n₁ and n₂, and n₁ >n₀and n₂ >n₀.
 12. The single mode optical fiber of claim 1 wherein theindex of refraction of the central region is greater than the clad layerand is equal to or less than about 11% of the index of retraction ofsaid at least one annular region surrounding the central region.
 13. Asingle mode optical waveguide fiber comprising:a core region comprising,a central region having a radius a₀ of substantially constant refractiveindex n₀, at least one annular region having a radius a₁ surroundingsaid central region, the innermost annular region is adjacent saidcentral region and has minimum refractive index n_(i) and n_(i) >n₀ ; aclad layer surrounding said core region having refractive index n_(c);characterized in that the ratio a₀ /a₁ is in the range of 0.55 to 0.75and the total dispersion slope is less than about 0.075 ps/nm² -km. 14.The single mode optical waveguide fiber of claim 13 characterized inthat core radius is in the range of about 3.5 to 3.9 microns and thezero dispersion wavelength is substantially constant for about 5%variations from target of core radius or cut off wavelength.
 15. Thesingle mode optical waveguide fiber of claim 13 characterized in thatthe core radius is greater than about 3.4 microns and the zerodispersion wavelength is outside the range of about 1530 nm to 1565 nm.16. The single mode optical waveguide fiber of claim 13 characterized inthat the core radius is between 3.4 and 3.5 microns and the zerodispersion wavelength is in the range of about 1500 nm to 1530 nm.
 17. Asingle mode optical waveguide fiber comprising:a core region comprising,an axially symmetric central region having minimum refractive index n₀,an axially symmetric annular region adjacent said central region, havingminimum refractive index n₁, inner radius a_(i), and outer radius a_(o),wherein n₁ >n₀ and the ratio a_(i) /a_(o) is in the range 0.35 to 0.80,and; a clad layer surrounding said core having refractive index n_(c)and n₁ >n_(c).
 18. The single mode fiber of claim 17 wherein the ratioa_(i) /a_(o) is about 0.50.
 19. The single mode optical waveguide fiberof claim 17 wherein n₀ is substantially constant.
 20. The single modeoptical waveguide fiber of claim 19 wherein n₀ is substantially equal ton_(c).
 21. The single mode optical waveguide fiber of claim 20 whereinn₁ is substantially constant.
 22. The single mode optical waveguidefiber of claim 21 wherein n₁ is in the range of about 1.4700 to 1.4800.